When we first moved in here, this stuff was all over the yard. It didn't take much to kill, just pulling it out and then using Round Up on any stragglers. This went from a roar to a few little sprigs in a year. I still pull them out before they make berries, so they don't have the chance to go crazy.
I'm of two minds about Mulberry... they do make nice shady trees, can be cut back into topiaries, and as long as they're far enough from something they can foul, such as a house or driveway - you don't want to see what the rotting berries will do to a hard surface - they can be nice. The berries are delicious and it can be very prolific. They're easy to kill, just cut them to the ground and they don't generally come back. Pull out or cut down the sprouts when you find them.
Tree of Heaven
Another fast growing, if truly foul tree. I cut these out whenever I see them, and a little round up on the trunks prevents them from coming back.
This stuff is everywhere. It self-seeds very prolifically. It's tough to contain without being very dutiful about weeding it out where you don't want it. That said, it smells nice and will grow where other things won't, and provides some late-summer interest when everything else is dying off.
At first I considered this as being something deliberately planted, since the previous owners as gardeners tended to plant as deer-resistantly as possible. Before I learned what this was, I thought it was some kind of rhubarb, since the leaves had similar size and shape, but the stems never got as big as rhubarb. I leave this in a few places since it fills in dark spots nicely and chop off the burs before they drop.
This is the bane of my gardening life. The previous owner didn't do much to the yard, and that let this get a complete foothold. I've been battling this for three years - it is one of the most pernicious and invasive weeds around. It drops berries that are very fecund, spread by birds, dropped where they are, they grow insanely fast into trees which are very difficult to kill, and are some of the most allelopathic (natural herbicidal) things I've seen. If you don't nip this shit in the bud as hard as you can, it will kill everything else in your yard. Any larger (>1' width) trees I've had to gird and tried several different herbicides on the stump to prevent it from regrowing. The only thing I've found as a sure-kill has been Tricopyr, after a year of trying Round-Up and having to gird several of the bigger trees two and three times. I've gotten so far as removing all the tree-sized monsters and monthly round-up every single sprig I see coming up. I fear this is becoming a losing battle because the yards behind me have let several large buckthorns grow into trees that are persistently dropping berries into my yard. Terrible shit!I'm sorely tempted to go out one late night and use a hand saw to kill it all in the surrounding yards.
The first year we were here this had completely taken over our back yard gardens. It's been pretty easy to kill with persistent ripping them out and using Round-Up on bigger clusters. This is probably the first one I've seen in two years.
This has been hovering at the edge of a wooded area in my yard and I've just resigned to watch it - it doesn't seem especially invasive but it has been forming pretty dense clusters.
(likely Glechoma hederacea)
This is everywhere. I could go on an herbicidal tear getting rid of this, but I decided to run an experiment instead - overplant white clover everywhere in the lawn and let the clover and everything else duke it out.
This is a new one in my yard - several large clusters of this popped up this year. I seem to recall planting it the first year we were here, but then didn't notice any growing. This year it went gangbusters! Gilligan has been thrilled.
The other thing that's generally considered a weed that I've enlisted to help is White Dutch Clover. I've gotten past the whole 'perfect lawn' lie and want my yard to be much more resilient than the monoculture of fescue. The last two years I've overplanted 10 lbs of white clover, and this year it's really starting to show. My grass is greener than it's ever been and stays lush even in the rain doldrums usual around late July, early August. I'd inoculated the clover with the bacillus they offered and the clover has shown up in all parts of the yard.
In the very back of our yard there's a somewhat old Ash tree that had mulch applied under it in a 10 yard circle, and that mulch had lost its effectiveness and was succumbing to weeds. Three years ago, I cut down the weeds as hard as I could, raked up and discarded what was left of the mulch, disturbed the dirt with a rake and seeded it heavily with white dutch clover. It completely took over and overcrowded out all of the remaining weeds, and that experiment led me to try overplanting it in the rest of the yard's lawn. There's a school of thought that the nitrogen fixation characteristic of clover will reduce the nitrogen hunger of normal grass. So far this year, the second year of overplanting it, that school seems to achieved its goal - I put down no fertilizer and the grass is greener, although that also partly owes to the leaves of the much-hardier clover being much more prominent. I kinda love it so far. The clover never grows so tall as to shade the grass, and it's been very tolerant of weekly mowing.